Find the Laplace transform of the function H(t−a)t^n

klastiesym

klastiesym

Answered question

2022-10-24

Find the Laplace transform of the function H ( t a ) t n . That's what I have done so far:
L { H ( t a ) t n } = 0 e s t H ( t a ) t n d t = a e s t t n d t = 0 e s ( t + a ) ( t + a ) n d t
Could you tell me how I can continue?

Answer & Explanation

t5an1izqp

t5an1izqp

Beginner2022-10-25Added 13 answers

You can obtain directly from the integral
L { H ( t a ) t n } = a e s t t n d t = 1 s n + 1 a s e z z n d z = 1 s n + 1 Γ ( n + 1 , a s )
recalling the integral expression of incomplete Gamma function for integer n
Γ ( n , x ) = x e z z n 1 d z .
NOTE Another useful methodology may be the following.
Observing that
H ( t a ) t n = H ( t a ) ( t a + a ) n = H ( t a ) k = 0 n ( n k ) ( t a ) k a n k
the Laplace transform becomes
L { H ( t a ) t n } = k = 0 n ( n k ) a n k L { H ( t a ) ( t a ) k }
recalling that L { H ( t a ) f ( t a ) } = e a s F ( s ). Using L { t n } = n ! s n + 1 , we obtain
L { H ( t a ) t n } = e a s k = 0 n ( n k ) a n k k ! s k + 1 = n ! e a s s n + 1 k = 0 n ( a s ) n k ( n k ) ! = 1 s n + 1 n ! e a s ν = 0 n ( a s ) ν ν !
and recalling that
Γ ( n , z ) = ( n 1 ) ! e z ν = 0 n 1 z ν ν !
is the incomplete Gamma function for integer n, we obtain finally
L { H ( t a ) t n } = 1 s n + 1 Γ ( n + 1 , a s ) .
Pellagra3d

Pellagra3d

Beginner2022-10-26Added 1 answers

You know that L { U ( t a ) } = e a s s , a > 0. And
L { t n f ( t ) } = ( 1 ) n ( L ( f ( t ) ) ) ( n )
Now let think about ( e a s s ) ( n ) . Using Maple, I found:
n = 1 ( e a s s ) = exp ( a s ) a s + 1 s 2 n = 2 ( e a s s ) = exp ( a s ) a 2 s 2 + 2 a s + 2 s 3 n = 3 ( e a s s ) = exp ( a s ) a 3 s 3 + 3 a 2 s 2 + 6 a s + 6 s 4
And so I think the possible pattern could be
( 1 ) n k = 0 n n ! ( n k ) ! s n k a n k s n + 1

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?